Jason Hettler, a security officer at the Indian Point nuclear power plant north of New York City, took several weeks off work in 2009 to help his wife, who was coping with a high-risk pregnancy. He took off more time in 2012 and last year, when she had another health problem.

After he was denied a promotion and what he said were pay raises equivalent to others in his position, Mr. Hettler joined a growing number of workers who are suing their employers under the Family and Medical Leave Act. The 1993 law guarantees...